Transforms

There is also the Decepticon matrix that transform any object to be Evil

Transforms are ways to transform coordinate spaces with matrices. The most common ones is scale, rotation and translation aka as SRT. To support the translation use a matrix one order larger then the dimension you work in. So in 2D a 3x3 matrix is used and in 3D a 4x4 matrix. The matrix is a grid of numbers like this.

Xx Xy Xz 0
Yx Yy Yz 0
Zx Zy Zz 0
Ox Oy Oz 1

In the matrix the (Ox, Oy, Oz) is the origin or position of the object. The rotation is given by the three base axis, (Xx, Xy, Xz) is the X vector, (Yx, Yy, Yz) is the Y vector and (Zx, Zy, zz) is the Z vector. It is common to call the X the 'right side', Y for 'up' and Z is 'forward'. When a position is multiplied with the transform the position will be transformed into transforms coordinate system or space.

Matrix Transformations - 2019

Affine transformations - 2018

Change of basis in Linear Algebra - 2015

Visualizing matrix multiplication as a linear combination - 2015

Staring Into The Matrix - 2014

Numbers in Transformation Matrices - 2012

Matrix Layouts, DirectX and OpenGL

The Matrix and Quaternions FAQ

Matrix Representation of Transformations

Transforming Objects using Matrices

Space

A vector space is number of linear independent (base) vectors. In 3D there are three of these and they can be scaled and added to obtain all other vectors in the space.

Why Clip Transformed Vertices in Clip-Space? - 2018

World, View and Projection Transformation Matrices - 2013

Model Space

World Space

View Space / Camera Space

Projection Space

Transforms

To perform transformations between different spaces matrices are used. These are the standard names for the matrices that transform between certain vector spaces. For a matrix that only contain SRT operations one can make a inverse matrix that transform the opposite way.

Model Matrix : Model Space -> World Space

View Matrix : World Space -> View Space

Model View Projection - 2019

Understanding the View Matrix - 2011

Projection Matrix : View Space -> Projection Space

The two projections are perspective and orthographic.

Inverse Transform

Transform direction/position/vector

Reference

Axonometric Projections - A Technical Overview - 2013

Mouse Picking Demystified - 2005

The Normal Equation and matrix calculus - 2015